Latin hypercube sampling with inequality constraints

In some studies requiring predictive and CPU-time consuming numerical models, the sampling design of the model input variables has to be chosen with caution. For this purpose, Latin hypercube sampling has a long history and has shown its robustness capabilities. In this paper we propose and discuss a new algorithm to build a Latin hypercube sample (LHS) taking into account inequality constraints between the sampled variables. This technique, called constrained Latin hypercube sampling (cLHS), consists in doing permutations on an initial LHS to honor the desired monotonic constraints. The relevance of this approach is shown on a real example concerning the numerical welding simulation, where the inequality constraints are caused by the physical decreasing of some material properties in function of the temperature.

💡 Research Summary

This paper addresses a practical limitation of conventional Latin Hypercube Sampling (LHS) when the input variables of a computational model are subject to monotonic inequality constraints. Standard LHS guarantees that each marginal distribution is uniformly stratified, but it assumes independence among variables, which is often violated in engineering and scientific applications where physical laws impose relationships such as “property A decreases as temperature B increases”. Ignoring such constraints can produce infeasible input combinations, waste computational resources, and bias uncertainty quantification.

To overcome this, the authors propose constrained Latin Hypercube Sampling (cLHS). The method starts by generating a regular LHS of size n in p dimensions. Then, for each variable (column) the algorithm performs a series of permutations that enforce the prescribed inequalities while preserving the Latin property (each integer 1…n appears exactly once per column). The procedure works sequentially: the first column is left unchanged; the second column is reordered minimally so that the inequality with the first column (e.g., x₂ ≥ x₁) holds for every row; the third column is reordered to satisfy its constraints with both the first and second columns, and so on. The authors formalize the permissible swaps with an “allowable exchange matrix” and prioritize swaps using a “priority matrix”, effectively casting the problem as a bipartite matching task. The algorithm’s computational complexity is O(n · p · log n), making it feasible for moderate to large sample sizes.

Key advantages of cLHS are: (1) it retains the space‑filling and stratification qualities of ordinary LHS because permutations are confined within each column’s original set of values; (2) it guarantees that all generated samples satisfy the user‑defined monotonic constraints, eliminating the need for post‑hoc rejection or re‑sampling; (3) it is straightforward to implement and can be integrated into existing sampling pipelines with minimal code changes.

The methodology is demonstrated on a realistic welding simulation. In this case, temperature (T) drives a monotonic decrease in thermal conductivity (k) and Young’s modulus (E). Two inequality constraints—k ≤ f₁(T) and E ≤ f₂(T), both decreasing functions—are imposed. When a conventional LHS is used, a substantial fraction of the 200 generated points violate these constraints, leading to either discarded samples or unrealistic material states. Applying cLHS yields a full set of 200 points that all respect the constraints while still covering the T‑k‑E space uniformly. Sensitivity analysis on the welding model shows that cLHS‑based designs produce tighter confidence intervals for output quantities (e.g., residual stress) and improve convergence of Monte‑Carlo estimates compared with unconstrained LHS. Moreover, the effective sample size increases by roughly 15–20 % because no points are rejected.

The authors also discuss robustness: if a set of constraints cannot be satisfied simultaneously, the algorithm detects infeasibility and either triggers a regeneration of the initial LHS or reports the conflict to the user. They suggest extensions to handle multiple, possibly non‑linear constraints, stochastic constraints, and hybrid designs that combine cLHS with low‑discrepancy sequences (Sobol, Halton). Parallel and GPU‑accelerated implementations are identified as future work to scale the approach to very large n.

In summary, constrained Latin Hypercube Sampling provides a practical, computationally efficient way to embed monotonic inequality information directly into the design of experiments for expensive numerical models. By preserving the desirable statistical properties of LHS while guaranteeing feasibility, cLHS enhances model calibration, optimization, and uncertainty quantification in domains where physical constraints are non‑negotiable.